The pair clashed as they left the field at half-time during the clash - which Spurs won 1-0 - but Mourinho felt it was necessary for the team after their recent defeat to Sheffield United.
The 57-year-old coach said: "(The argument) was probably as a consequence of our meetings. If you want to blame someone for that it's me.
"I was critical of my boys. They were not critical enough with themselves. I asked them to be more demanding of each other.
"Son is an amazing kid. Everybody likes Son, but the captain told him you have to do more and give more to the team.
"It is something needed for the team to grow up. To grow up you need big personalities. When you have that reaction I have no doubts."
Lloris admitted the incident, where he reacted furiously after Son didn't track back for an Everton attack, "belongs to the changing room".
The World Cup-winning goalkeeper said: "It just belongs in the changing room - outside you can say whatever you want. What happened between me and Sonny is part of football sometimes.
"At the end of the game you could see we were more than happy."
A Michael Keane own goal gave Tottenham victory on Monday night (06.07.20) and saw Mourinho record his 200th Premier League victory, becoming only the second manager to do so after Sir Alex Ferguson.
Discussing the landmark, the former Chelsea and Manchester United manager said: "Not many managers work in three of the top football clubs in the country, and in the world.
"I was lucky enough to work for three of the biggest clubs in the country. When you are one of those big clubs it's easier to have these results."